Bayesian survival analysis using host type as a predictor

library(rstan)
library(survival)
library(tidyverse)
library(tidybayes)
library(scales)
library(survminer)

# data, parameters, model and generated quantities blocks
Stan_exponential_survival_model <- "
data{
  int <lower=1> N_uncensored;
  int <lower=1> N_censored;
  int <lower=0> numCovariates;
  matrix[N_censored, numCovariates] X_censored;
  matrix[N_uncensored, numCovariates] X_uncensored;
  vector <lower=0>[N_censored] times_censored;
  vector <lower=0>[N_uncensored] times_uncensored;
}

parameters{
  vector[numCovariates] beta; //regression coefficients
  real alpha; //intercept
}

model{
  beta ~ normal(0,10); //prior on regression coefficients
  alpha ~ normal(0,10); //prior on intercept
  target += exponential_lpdf(times_uncensored | exp(alpha+X_uncensored * beta)); //log-likelihood part for uncensored times
  target += exponential_lccdf(times_censored | exp(alpha+X_censored * beta)); //log-likelihood for censored times
}

generated quantities{
  vector[N_uncensored] times_uncensored_sampled; //prediction of death
  for(i in 1:N_uncensored) {
    times_uncensored_sampled[i] = exponential_rng(exp(alpha+X_uncensored[i,]* beta));
  }
}
"

# prepare the data
set.seed(42); 
require (tidyverse);
data <- read_csv('../data/necessary_fields.csv')

Rows: 2066 Columns: 7
── Column specification ───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Delimiter: ","
chr (1): host_type
dbl (1): duration_months
lgl (5): major_releases, censored, high_rev_freq, multi_repo, high_author_count

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

data$host_type <- car::recode(data$multi_repo, "'git' = 0; 'pypi' = 1; 'deb' = 2")
data$censored <- car::recode(data$censored, "'TRUE' = 0; 'FALSE' = 1")
N <- nrow (data); 
X <- as.matrix(pull(data, host_type)); 
is_censored <- pull(data, censored)==0; 
times <- pull(data, duration_months); 
msk_censored <- is_censored == 1; 
N_censored <- sum(msk_censored);

# put data into a list for Stan
Stan_data <- list (N_uncensored = N - N_censored, 
                    N_censored = N_censored,
                    numCovariates = ncol(X), 
                    X_censored = as.matrix(X[msk_censored,]),
                    X_uncensored = as.matrix(X[!msk_censored ,]), 
                    times_censored = times[msk_censored],
                    times_uncensored = times[!msk_censored])

# fit Stan model
require(rstan)
exp_surv_model_fit <- suppressMessages(stan(model_code = Stan_exponential_survival_model, data = Stan_data))

sh: Data/bayesian: No such file or directory
Warning in system2(CXX, args = ARGS) : error in running command
sh: clang++ -mmacosx-version-min=10.13: command not found
Warning in file.remove(c(unprocessed, processed)) :
  cannot remove file '/var/folders/q8/7tchbyvd1dj3hkgw5ffkk6ph0000gp/T//RtmpKsHNQc/file9c706c9c0cef.stan', reason 'No such file or directory'

SAMPLING FOR MODEL 'bf5dbbde6a245330de71a285e3fe7c42' NOW (CHAIN 1).
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SAMPLING FOR MODEL 'bf5dbbde6a245330de71a285e3fe7c42' NOW (CHAIN 2).
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SAMPLING FOR MODEL 'bf5dbbde6a245330de71a285e3fe7c42' NOW (CHAIN 3).
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# print model fit
print(get_seed(exp_surv_model_fit))

[1] 1781592037

# print fit summary
fit_summary <- summary(exp_surv_model_fit)
print(fit_summary$summary)

                                      mean     se_mean           sd         2.5%          25%          50%          75%         97.5%    n_eff      Rhat
beta[1]                          -1.241249 0.004399075   0.21966967    -1.704852    -1.385844    -1.227850    -1.090840    -0.8365519 2493.547 1.0006737
alpha                            -4.760864 0.000692821   0.03681899    -4.832656    -4.785862    -4.760762    -4.735189    -4.6901059 2824.241 1.0021146
times_uncensored_sampled[1]     116.110950 1.816253820 113.89904023     3.828090    35.552880    81.160012   160.792540   416.9041709 3932.666 1.0003167
times_uncensored_sampled[2]     117.398403 1.918345271 119.68237121     3.085864    32.699468    80.545219   161.571267   435.2324032 3892.305 1.0004681
times_uncensored_sampled[3]     116.758176 1.810254034 117.60818038     3.536132    33.567905    81.820927   160.187807   437.5175025 4220.812 1.0013378
times_uncensored_sampled[4]     112.577890 1.784403266 110.48722921     2.563546    33.320063    80.184923   154.958987   412.2973216 3833.877 1.0000911
times_uncensored_sampled[5]     114.618238 1.832611854 116.78313654     3.625778    32.521434    79.563019   157.571901   430.6085149 4060.872 0.9994195
times_uncensored_sampled[6]     117.038422 1.879269314 117.82323367     3.254252    33.732012    80.502100   160.567931   443.3927199 3930.826 0.9994286
times_uncensored_sampled[7]     116.343737 1.829307595 117.35987530     2.991287    33.772248    80.131601   162.067532   433.9440970 4115.909 1.0007248
times_uncensored_sampled[8]     117.337301 1.871982416 119.73311099     3.082806    34.411068    79.122300   158.867584   448.1908723 4090.958 1.0000238
times_uncensored_sampled[9]     113.576107 1.794125104 112.22040585     2.596768    32.890424    78.825406   158.343305   411.6722516 3912.355 1.0014255
times_uncensored_sampled[10]    120.619010 1.904630791 120.68967175     3.213133    34.523374    84.593657   167.649775   427.9242035 4015.306 0.9995474
times_uncensored_sampled[11]    116.389529 1.951866343 119.98003807     2.982066    32.119019    77.843162   160.121067   448.0154818 3778.486 1.0007529
times_uncensored_sampled[12]    116.204652 1.840418684 116.59452920     2.553634    34.302267    80.677522   163.428421   427.6304305 4013.498 1.0001575
times_uncensored_sampled[13]    118.694306 1.923276426 120.90594812     2.590976    34.346900    82.690796   164.204956   435.3176321 3951.954 1.0010909
times_uncensored_sampled[14]    118.444990 1.821757502 118.71940555     2.721791    35.263761    82.287013   162.854422   432.8318950 4246.805 1.0000885
times_uncensored_sampled[15]    113.595537 1.890307098 114.29974546     3.053854    33.955957    77.770006   156.699720   420.3014283 3656.165 0.9999990
times_uncensored_sampled[16]    115.967457 1.883735054 116.96980625     3.545682    33.806978    78.435021   160.029956   429.7512941 3855.741 0.9997824
times_uncensored_sampled[17]    115.587196 1.847963118 116.81433296     2.572873    33.090329    77.417018   162.841511   419.4268461 3995.818 0.9997818
times_uncensored_sampled[18]    117.126859 1.853295053 118.29038089     3.216291    33.429437    80.618084   163.280868   426.0217048 4073.894 0.9996100
times_uncensored_sampled[19]    118.165339 1.901142754 117.87694199     2.922210    34.002676    79.778306   164.878472   424.7693566 3844.397 1.0007652
times_uncensored_sampled[20]    118.734930 1.879253246 118.35473039     3.057919    33.830442    82.453984   163.764279   449.2054989 3966.438 1.0006368
times_uncensored_sampled[21]    117.143133 1.842994778 118.10512338     2.680719    32.457869    78.617016   161.881827   429.8791671 4106.665 1.0001425
times_uncensored_sampled[22]    116.500056 1.857142823 118.50871119     2.943743    32.265244    78.986055   159.071715   429.2813895 4072.020 0.9997388
times_uncensored_sampled[23]    115.920159 1.782347849 115.01141574     2.444619    33.382755    80.940287   160.150071   431.4323819 4163.868 1.0002382
times_uncensored_sampled[24]    402.559014 6.846595810 411.86460271     9.400620   113.762385   281.906768   550.040667  1530.1814643 3618.758 0.9997602
times_uncensored_sampled[25]    114.218399 1.817493876 112.46959805     3.234372    32.870748    81.413978   158.424384   410.5547879 3829.344 1.0002147
times_uncensored_sampled[26]    116.165190 1.803635523 116.60003427     2.782752    32.866759    80.130920   161.463704   440.5737721 4179.264 0.9994917
times_uncensored_sampled[27]    118.185894 1.838761904 116.32042565     3.477241    34.754141    83.466567   162.859867   440.3312296 4001.852 0.9994571
times_uncensored_sampled[28]    114.981472 1.817927282 113.83248519     3.159084    33.519241    78.915896   159.951172   417.8982752 3920.843 0.9994676
times_uncensored_sampled[29]    117.263579 1.760709098 111.88870588     2.830289    35.839522    83.644769   163.764710   417.7855585 4038.288 0.9994909
times_uncensored_sampled[30]    117.096958 1.872004272 115.04188786     3.420780    33.387420    81.852590   162.132437   425.3680083 3776.577 0.9999145
times_uncensored_sampled[31]    116.142803 1.871447446 115.19633594     3.591958    33.575190    79.709210   158.917945   418.6409822 3788.978 1.0002945
times_uncensored_sampled[32]    114.812943 1.857392033 113.67697615     2.721071    32.724348    79.575512   159.489724   425.0519014 3745.742 1.0001875
times_uncensored_sampled[33]    114.522020 1.814291777 115.25853879     2.904442    32.469960    78.904497   158.935318   424.7760882 4035.822 0.9996374
times_uncensored_sampled[34]    115.764148 1.886369449 118.32228708     2.847562    33.251455    78.925981   158.172809   435.8701544 3934.410 1.0003886
times_uncensored_sampled[35]    115.983168 1.914025260 116.47865633     2.942893    34.700379    78.857795   158.822943   427.2183141 3703.372 1.0000415
times_uncensored_sampled[36]    113.520485 1.843405423 115.14388862     2.922958    31.553842    79.454802   156.757565   431.9367504 3901.576 1.0002835
times_uncensored_sampled[37]    115.493064 1.831955866 115.88584120     2.932132    31.470484    79.318212   162.608548   434.6768018 4001.573 0.9998782
times_uncensored_sampled[38]    117.393111 1.830815827 116.41761055     2.678494    33.078180    81.505803   162.500631   441.1332552 4043.412 0.9994226
times_uncensored_sampled[39]    116.798690 1.826460689 112.31070107     3.605440    35.397505    83.511888   162.447592   416.9615009 3781.130 1.0001357
times_uncensored_sampled[40]    117.289976 1.807921232 116.40765875     3.072815    35.612128    82.807961   160.871615   422.3483814 4145.759 0.9999597
times_uncensored_sampled[41]    115.655103 1.862746206 116.42877732     2.759386    33.165181    81.186780   159.239227   437.1524438 3906.729 1.0005037
times_uncensored_sampled[42]    117.064895 1.915011647 117.49049018     3.075755    33.100800    81.582581   161.455660   425.1218341 3764.112 1.0002698
times_uncensored_sampled[43]    116.578686 1.867450669 116.65348838     2.794526    32.438795    80.942766   162.204124   437.2413140 3902.089 1.0000977
times_uncensored_sampled[44]    116.826451 1.856359331 119.67671218     2.643116    32.961708    79.376293   161.225113   430.0583106 4156.188 0.9996589
times_uncensored_sampled[45]    113.875257 1.854911740 115.46043172     3.090699    32.876240    77.948552   153.822782   434.3188533 3874.537 0.9994250
times_uncensored_sampled[46]    118.164114 1.976833641 118.86469308     2.946345    33.325162    78.964820   166.696730   450.3527934 3615.476 1.0004055
times_uncensored_sampled[47]    114.350041 1.950981030 118.69704670     2.518240    31.787881    77.398433   157.237492   434.4216503 3701.466 0.9996946
times_uncensored_sampled[48]    115.329410 2.032661809 116.65837408     2.527090    31.138967    80.469633   161.172436   425.8236743 3293.833 0.9998481
times_uncensored_sampled[49]    118.675544 1.855547493 116.31824157     3.337705    35.822523    84.910738   167.909527   409.0140169 3929.629 0.9997802
times_uncensored_sampled[50]    116.193673 1.800378781 117.08932651     2.986716    33.428339    81.438114   159.364798   435.8131308 4229.674 1.0000285
times_uncensored_sampled[51]    117.046782 1.815935526 116.03935940     3.187660    33.477695    81.133266   164.490428   429.5914366 4083.286 0.9994440
times_uncensored_sampled[52]    115.535379 1.897806525 118.35645121     3.175386    30.979655    77.833070   159.781504   428.1984418 3889.377 1.0000948
times_uncensored_sampled[53]    115.512411 1.870600596 117.49123220     2.434449    32.363888    80.373686   157.081654   430.0283729 3945.016 0.9998184
times_uncensored_sampled[54]    118.845608 1.965303036 119.97379005     2.584567    33.139854    82.013101   165.596792   438.5357311 3726.608 1.0002825
times_uncensored_sampled[55]    114.538468 1.807177168 117.42571971     2.923251    33.079270    79.427039   157.784305   443.1473109 4222.066 0.9994917
times_uncensored_sampled[56]    118.909049 1.882464435 116.43094210     3.024994    35.851011    83.387558   166.613053   422.9086638 3825.456 1.0000517
times_uncensored_sampled[57]    118.559533 1.834599361 118.20326408     3.081301    33.106766    84.809784   164.774580   432.4123305 4151.227 1.0001961
times_uncensored_sampled[58]    114.672828 1.877437282 116.40287230     3.069028    33.168786    76.178358   157.068600   431.3730015 3844.116 0.9999382
times_uncensored_sampled[59]    116.008385 1.801996099 115.14431857     2.902354    33.577559    80.793998   163.762867   416.3843519 4082.981 1.0005042
times_uncensored_sampled[60]    116.092998 1.921733221 116.37596459     3.190643    33.687861    80.516086   159.266380   424.1321353 3667.249 0.9991897
times_uncensored_sampled[61]    118.903273 1.967525931 119.37287933     3.383155    34.962019    81.370075   165.475695   452.3111627 3681.039 1.0011655
times_uncensored_sampled[62]    116.299242 1.808002805 115.31697668     2.800810    33.119073    79.946455   161.927035   426.8520484 4068.069 0.9997926
times_uncensored_sampled[63]    119.184066 1.851028740 120.98972283     2.757871    34.825022    83.181642   159.690195   446.0108775 4272.388 0.9996026
times_uncensored_sampled[64]    118.054292 2.015792959 118.80670188     2.619900    35.030999    83.142496   160.050854   444.5048045 3473.682 1.0009968
times_uncensored_sampled[65]    116.057373 1.889453714 117.89748411     2.278490    32.738474    78.541151   161.230776   439.9083995 3893.468 1.0000098
times_uncensored_sampled[66]    118.758672 1.890580081 118.56026241     3.209833    33.404673    82.819860   166.798541   427.4084297 3932.676 0.9996844
times_uncensored_sampled[67]    116.707900 1.882361287 117.07228992     2.923595    34.045457    80.931718   162.188247   416.5580470 3868.141 1.0003770
times_uncensored_sampled[68]    115.529232 1.921982328 119.22575035     2.790840    31.753396    79.719858   156.640127   441.3412662 3848.056 0.9994784
times_uncensored_sampled[69]    117.866718 1.959546017 122.27354747     2.543300    32.151047    81.112942   159.661701   452.0983387 3893.625 0.9995756
times_uncensored_sampled[70]    114.411277 1.900279806 117.10331961     2.842611    32.680012    78.119869   155.754257   434.1195360 3797.548 1.0008711
times_uncensored_sampled[71]    113.184351 1.758093188 113.19705392     2.879391    31.372438    77.802280   157.347418   428.6998820 4145.591 0.9997317
times_uncensored_sampled[72]    117.855704 1.978819767 120.35792850     2.720176    33.543771    80.853092   162.507536   441.9170345 3699.448 0.9995937
times_uncensored_sampled[73]    119.005249 1.796975494 115.01623835     3.134401    35.140565    84.530000   166.276546   424.4764389 4096.699 0.9996389
times_uncensored_sampled[74]    115.605314 1.785641301 114.58430644     2.799568    33.879160    80.069025   159.122499   427.9012755 4117.768 0.9998759
times_uncensored_sampled[75]    116.310909 2.007751845 121.59880641     2.773391    31.792354    78.352622   162.990097   437.8835685 3668.078 0.9991733
times_uncensored_sampled[76]    117.688241 1.832442459 117.56774192     3.286352    33.706103    80.466626   166.339376   429.9904769 4116.382 0.9997498
times_uncensored_sampled[77]    120.583356 1.871002472 119.77126660     2.711602    34.652339    83.315014   170.018778   446.7144501 4097.855 1.0005092
times_uncensored_sampled[78]    115.048227 1.968745850 116.16408095     2.746027    32.476688    80.353153   157.776691   433.9112169 3481.484 1.0001414
times_uncensored_sampled[79]    116.015936 1.927863892 117.60789384     2.706462    33.491197    79.406487   159.494906   442.0551743 3721.519 1.0000646
times_uncensored_sampled[80]    117.156073 1.917467228 122.14464093     3.029535    32.753171    77.576765   159.713403   430.1114832 4057.821 1.0002067
times_uncensored_sampled[81]    116.534914 1.907849838 115.77992478     2.854398    34.556736    81.870737   162.733509   415.1151629 3682.800 0.9997981
times_uncensored_sampled[82]    114.998134 1.783946162 115.08081152     3.244805    33.219098    81.027483   161.247642   432.1029483 4161.428 0.9993863
times_uncensored_sampled[83]    113.429676 1.851548312 114.43483756     3.353587    33.413956    77.892486   155.759961   416.5879755 3819.851 0.9997041
times_uncensored_sampled[84]    122.316666 1.920223291 119.55968794     3.330546    34.857091    85.563775   171.866609   440.1484439 3876.735 0.9998309
times_uncensored_sampled[85]    117.918289 1.905093904 118.43025840     2.684521    33.612832    81.891795   164.670333   435.9835082 3864.493 1.0004106
times_uncensored_sampled[86]    115.119372 1.826034496 113.63800032     3.365266    32.003802    80.419011   161.482858   419.5317642 3872.837 1.0001156
times_uncensored_sampled[87]    121.535003 2.018805534 123.89644337     2.832316    34.946796    83.406417   165.600818   445.1002628 3766.420 0.9997706
times_uncensored_sampled[88]    117.230999 1.953752753 116.15974768     3.336650    34.811422    82.158188   163.429734   444.7335154 3534.859 1.0011126
times_uncensored_sampled[89]    113.807382 1.881188160 115.07186464     2.966997    33.026017    78.305410   156.670341   428.6233507 3741.742 0.9996693
times_uncensored_sampled[90]    119.295605 1.816445422 115.60206430     2.845775    34.688566    83.622052   168.156164   426.3435433 4050.293 1.0004213
times_uncensored_sampled[91]    116.161160 1.948668494 118.54732649     2.763000    32.240629    80.631102   161.006432   430.8492728 3700.902 0.9997977
times_uncensored_sampled[92]    120.046034 1.985813626 121.78704729     3.326654    34.481540    81.980546   165.689458   451.5202064 3761.190 0.9996364
times_uncensored_sampled[93]    115.811082 1.867445120 115.30698882     2.993639    33.545040    81.044043   161.125355   424.2905367 3812.550 0.9996196
times_uncensored_sampled[94]    119.178293 1.819733729 116.22041166     3.384498    35.396355    81.927012   168.223854   435.8208637 4078.957 0.9999769
times_uncensored_sampled[95]    117.330917 1.862728438 116.99255981     2.292318    32.929276    81.880220   163.814856   433.6480886 3944.731 0.9997836
times_uncensored_sampled[96]    118.350554 1.884188438 119.48825007     2.998990    34.714395    82.092479   163.080401   441.4451897 4021.626 0.9997336
times_uncensored_sampled[97]    119.161283 2.035556764 121.09415520     2.983999    33.530977    82.648493   165.852193   445.3428833 3538.995 0.9999382
times_uncensored_sampled[98]    114.665469 1.827699470 114.12517207     2.968853    35.213496    81.172989   160.427453   414.8588777 3899.001 1.0014504
 [ reached getOption("max.print") -- omitted 678 rows ]

exp_surv_model_draws <- tidybayes::tidy_draws(exp_surv_model_fit)
exp_surv_model_draws

## Constructor for Strata-specific survival function
construct_survival_function <- function(alpha, beta, x) {
    function(t) {
        lambda <- exp(alpha + x*beta)
        exp(-(lambda * t))
    }
}

## Random functions
exp_surv_model_surv_func <-
    exp_surv_model_draws %>%
    select(.chain, .iteration, .draw, alpha, `beta[1]`) %>%
    ## Simplify name
    rename(beta = `beta[1]`) %>%
    ## Construct realization of random functions
    mutate(`S(t|2)` = pmap(list(alpha, beta), function(a,b) {construct_survival_function(a,b,2)}),
           `S(t|1)` = pmap(list(alpha, beta), function(a,b) {construct_survival_function(a,b,1)}),
           `S(t|0)` = pmap(list(alpha, beta), function(a,b) {construct_survival_function(a,b,0)}))
exp_surv_model_surv_func

times <- seq(from = 0, to = 165, by = 0.1)
times_df <- data_frame(t = times)

## Try first realizations
exp_surv_model_surv_func$`S(t|2)`[[1]](times[1:10])

 [1] 1.0000000 0.9999552 0.9999105 0.9998657 0.9998210 0.9997762 0.9997315 0.9996867 0.9996420 0.9995972

exp_surv_model_surv_func$`S(t|1)`[[1]](times[1:10])

 [1] 1.0000000 0.9998003 0.9996006 0.9994009 0.9992013 0.9990018 0.9988022 0.9986027 0.9984033 0.9982039

exp_surv_model_surv_func$`S(t|0)`[[1]](times[1:10])

 [1] 1.0000000 0.9991090 0.9982189 0.9973295 0.9964409 0.9955531 0.9946661 0.9937799 0.9928945 0.9920099

## Apply all realizations
exp_surv_model_survival <-
    exp_surv_model_surv_func %>%
    mutate(times_df = list(times_df)) %>%
    mutate(times_df = pmap(list(times_df, `S(t|2)`, `S(t|1)`, `S(t|0)`),
                           function(df, s2, s1, s0) {df %>% mutate(  s2 = s2(t),
                                                               s1 = s1(t),
                                                               s0 = s0(t))})) %>%
    select(-`S(t|2)`, -`S(t|1)`, -`S(t|0)`) %>%
    unnest(cols = c(times_df)) %>%
    gather(key = Strata, value = survival, s2, s1, s0) %>%
    mutate(Strata = factor(Strata, # Strata is for different host types
                              levels = c("s2","s1","s0"),
                              labels = c("host type=Git","host type=Pypi","host type=Debian")))

## Average on survival scale
exp_surv_model_survival_mean <-
    exp_surv_model_survival %>%
    group_by(Strata, t) %>%
    summarize(survival_mean = mean(survival),
              survival_95upper = quantile(survival, probs = 0.975),
              survival_95lower = quantile(survival, probs = 0.025))

`summarise()` has grouped output by 'Strata'. You can override using the `.groups` argument.

exp_surv_model_survival

# plot the graphs
(ggplot(data = exp_surv_model_survival, mapping = aes(x = t, y = survival, color = Strata, group = interaction(.chain,.draw,Strata))) 
  + theme_survminer()
 + geom_line(size = 0.1, alpha = 0.02) 
 + geom_line(data = exp_surv_model_survival_mean, mapping = aes(y = survival_mean, group = Strata)) 
 + geom_line(data = exp_surv_model_survival_mean, mapping = aes(y = survival_95upper, group = Strata), linetype = "dotted") 
 + geom_line(data = exp_surv_model_survival_mean, mapping = aes(y = survival_95lower, group = Strata), linetype = "dotted")
 + scale_y_continuous(limits=c(0,1))
 + scale_x_continuous(breaks = c(0,40,80,120,160))
 +labs(x = "Time (Months)", y = "Survival probability"))